Insurance: Mathematics and Economics 27 2000 113–122
Arithmetization of distributions and linear goal programming
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José L. Vilar
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Departamento de Economía Financiera y Actuarial, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Pabellón de 5 curso, Campus de Somosaguas, Pozuelo de Alarcón, Madrid 28223, Spain
Received 1 September 1998; received in revised form 1 December 1999; accepted 20 January 2000
Abstract
Linear goal programming can be used as a complementary technique when local moment matching method up to the second moment gives some negative mass. This could happen when manipulating a discrete or mixed type severity distribution. In
that case we can avoid a simple retreat to the first moment and look for an arithmetic distribution with equal mean and the second moment closest to that of the original distribution. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Arithmetic distribution; Local moment matching; Linear goal programming; Simplex method
1. Introduction
Arithmetization of probability distribution functions is often required when the user arrives to the numerical stage. This is specially the case when using Panjer’s algorithm Panjer and Willmot, 1992 , Corollary 6.6.1 in the numerical
evaluation of compound distributions with nonarithmetic severities, the latter being discrete with an irregular span, absolutely continuous or mixed types Panjer and Willmot, 1992, p. 223. This numerical evaluation can be handled
in the resolution of many problems as for example: approximating the probabilities of the distribution of total claims or some ruin probability in both the cases of classical continuous-time with infinite horizon and fixed-time ruin
probabilities, and finally, in the search of bounds or approximations of stop-loss transforms. Examples of these applications are found for instance in Gerber 1982, Gerber and Jones 1976, Panjer and Lutek 1983, Panjer and
Willmot 1992, and finally De Vylder 1996 where arithmetization is studied inside the topic of optimization.
Given a distribution function, there are many ways to look for an arithmetic equivalent. If this equivalence depends on the equality up to some order n ∈ {0, 1, 2, . . . } between the moments of the former and the latter, the available
techniques are grouped under the denomination of local moment matching methods. In this paper we focus on the major drawback of these methods, namely the possibility of obtaining negative masses when “arithmetizing” some
discrete or mixed type distribution for a given span h 0, and we propose a complementary approach based on the nearness of local moments. The method of the nearest second moment is built in terms of linear programming
more precisely in terms of linear goal programming. Our objective is to show that this technique may furnish more confident solutions than the ones produced by local moment matching.
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This work is granted by Direcci´on General de Enseñanza Superior, Ministerio de Educaci´on y Cultura, Project reference PB96-0099.
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Tel.: +34-91-3942578; fax: +34-91-3942570. E-mail address: econ103sis.ucm.es J.L. Vilar.
0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 0 0 0 0 0 4 0 - 8
114 J.L. Vilar Insurance: Mathematics and Economics 27 2000 113–122
2. Local moment matching methods
